Testing for harmonizability

Abstract

Let be a covariance function having the representation begin{equation} R(s,t) = int_{-infty}^{infty} int_{-infty}^{infty} exp (isx - ity)d^2 G(x,y) end{equation} where is continuous to the right in both variables and is of bounded variation in the plane; then is harmonizable in that is also a covariance. We give examples in which this result is used to determine the harmonizability of new processes and covariances that are formed by operations on old processes and covariances. Specifically, if is a real Gaussian harmonizable process, then is harmonizable. If is harmonizable, has bounded support and is a Fourier-Stieltjes transform, then and are harmonizable. If begin{equation} X(t) =int_{-infty}^{infty} f(t,u) dZ(u) end{equation} where is a Fourier-Stieltjes transform and has finite total variation, then is harmonizable. We also obtain a sufficient condition for Priestley's oscillatory processes to be harmonizable. We find that the Bochner-Eberlein characterization of Fourier-Stieltjes transforms, while not the only method, is particularly convenient for determining the harmonizability of these examples.